Der Mitarbeiter bittet uns, im Wartebereich leise zu bleiben.

Der Mitarbeiter is singular and masculine: the (male) employee / staff member.
Die Mitarbeiter would be plural: the employees.
(There’s also die Mitarbeiterin for a singular female employee.)

Uns is accusative. The verb bitten typically takes an accusative person:

• jemanden (Akk.) bitten = to ask someone
  So: Der Mitarbeiter bittet uns = The employee asks us.

Because im Wartebereich leise zu bleiben is an infinitive clause (an Infinitivgruppe) depending on bittet. German often uses a comma to separate such clauses, and with verbs like bitten it’s standard and very common.

It’s bitten + accusative object + infinitive clause with zu:

• jemanden bitten, etwas zu tun = to ask someone to do something
  Here, the “do something” part is (leise) zu bleiben (to stay quiet).

After verbs like bitten, German frequently uses an infinitive construction instead of dass:

• Natural: Er bittet uns, leise zu bleiben.
• Also possible but heavier: Er bittet darum, dass wir leise bleiben.
  The zu-infinitive is more compact and very typical in spoken and written German.

In a bitten + zu-infinitive structure, the implied subject of the infinitive is usually the object of bitten.
So uns is the ones who should leise bleiben.

Im is a contraction of in dem.
Wartebereich is masculine (der Wartebereich), so in dem Wartebereich → im Wartebereich.

Because in is a two-way preposition:

• Dative for location (where?): im Wartebereich = in the waiting area
• Accusative for direction (where to?): in den Wartebereich = into the waiting area
  Here it’s about being located there, so dative is used.

Leise is used as an adverb here (describing how you should remain):

• leise bleiben = to remain quiet
  Adverbs don’t take adjective endings in German.

In infinitive clauses, adverbs typically come before the infinitive verb:

• leise zu bleiben
  Putting leise after would sound wrong in standard German.

Bittet is the 3rd person singular present tense of bitten:

• ich bitte
• du bittest
• er/sie/es bittet
  So Der Mitarbeiter bittet = The employee asks.

You can, but the tone changes:

• bittet = asks / requests (polite, softer)
• fordert (uns) auf = instructs / tells (us) to (more direct, can sound stricter)
  Both can be used in similar situations, depending on how strong the request is.